Re: 1962 Evinrude rebuild
S&A,
It depends on the parts. Basics like waterpump kits, seal kits, carburetor kits, and ignition components may still be available at your local dealer, particularly if they've been in business since the Eisenhauer administration.
Parts like pistons, rings, rods, etc. are less likely to be found new, unless your local dealer happens to have a large stock of old-stock parts. There are some sources online like SeaWay Marine, Twin Cities Outboard, etc. that deal in vintage parts, but it can be hit and miss. You didn't mention which model engine you have, so it can vary depending on what you have, too.
If your motor has seen salt, which may explain the broken bolts, then it may be better in the long run to find another motor. If the broken bolt is a fluke, then you may just need to use modest amounts of heat (propane torch or heat gun) and penetrating oil to assure easier removal. If you're set on that particular motor for sentimental reasons, then you may need to find a parts motor to scavenge parts from. As you'll see in other posts on old motors, they're neither rare, nor expensive to acquire.
Bottom line, if the motor is as clean as you say, and is basically healthy as it is (good compression, shifts gears, includes controls, etc.) then it may not need much more than a tune-up to get it going reliably. If it is "blown", .ie a broken piston, low compression, noisy bearings, etc., then the parts motor route may be best...
Good luck!